Maximal independence and symmetry in crystal chemistry of natural tectosilicates.

Löwenstein's avoidance rule in aluminosilicates is reinterpreted on the basis of the fourth Pauling rule. It is shown that avoidance of Si-O-Si bridges may account for avoidance of Al-O-Al bridges. In view of this interpretation, it is proposed that the most favourable distributions of cations entering in substitution of silicon in the framework are associated to maximal independent sets of the respective 3-periodic nets.

Combinatorial aspects of the Löwenstein avoidance rule. Part III: the relational system of configurations.

This paper introduces a new method of determining the independence ratio of periodic nets, based on the observation that, in any maximum independent set of the whole net, be it periodic or not, the vertices of every unit cell should constitute an independent set, called here a configuration. For 1-periodic graphs, a configuration digraph represents possible sequences of configurations of the unit cell along the periodic line. It is shown that maximum independent sets of the periodic graph are based on directed cycles with the largest ratio.

Combinatorial aspects of the Löwenstein avoidance rule. Part II: local and global constraints.

The determination of the independence ratio of a periodic net requires finding a subgroup of the translation group of the net for which the quotient graph and a fundamental transversal have the same independence ratio; the respective motif defines a periodic factor of the net. This article deals with practical issues regarding the calculation of the independence ratio of mainly 2-periodic nets, with an application to the 200 2-periodic nets listed on the RCSR (Reticular Chemistry Structure Resource) site. A companion paper described a calculation technique of independence ratios of finite graphs based on propositional calculus.

Combinatorial aspects of the Löwenstein avoidance rule. Part I: the independence polynomial.

According to Löwenstein's rule, Al-O-Al bridges are forbidden in the aluminosilicate framework of zeolites. A graph-theoretical interpretation of the rule, based on the concept of independent sets, was proposed earlier. It was shown that one can apply the vector method to the associated periodic net and define a maximal Al/(Al+Si) ratio for any aluminosilicate framework following the rule; this ratio was called the independence ratio of the net.

Vertex collisions in 3-periodic nets of genus 4.

Unstable nets, by definition, display vertex collisions in any barycentric representation, among which are approximate models for the associated crystal structures. This means that different vertex lattices happen to superimpose when every vertex of a periodic net is located at the centre of gravity of its first neighbours. Non-crystallographic nets are known to be unstable, but crystallographic nets can also be unstable and general conditions for instability are not known.

Groupoids and labelled quotient graphs: a topological analysis of the modular structure in pyroxenes.

The analysis of the modular structure of pyroxenes, recently discussed in Nespolo & Aroyo [Eur. J. Mineral.

Vertex-connectivity in periodic graphs and underlying nets of crystal structures.

Periodic nets used to describe the combinatorial topology of crystal structures have been required to be 3-connected by some authors. A graph is n-connected when deletion of less than n vertices does not disconnect it. n-Connected graphs are a fortiari n-coordinated but the converse is not true.

Topological features in crystal structures: a quotient graph assisted analysis of underlying nets and their embeddings.

Topological properties of crystal structures may be analysed at different levels, depending on the representation and the topology that has been assigned to the crystal. Considered here is the combinatorial or bond topology of the structure, which is independent of its realization in space. Periodic nets representing one-dimensional complexes, or the associated graphs, characterize the skeleton of chemical bonds within the crystal.

Non-crystallographic nets: characterization and first steps towards a classification.

Non-crystallographic (NC) nets are periodic nets characterized by the existence of non-trivial bounded automorphisms. Such automorphisms cannot be associated with any crystallographic symmetry in realizations of the net by crystal structures. It is shown that bounded automorphisms of finite order form a normal subgroup F(N) of the automorphism group of NC nets (N, T).

Topological density of lattice nets.

It was shown in a previous paper [Eon (2004). Acta Cryst. A60, 7-18] that the topological density of a periodic net can be calculated directly from its cycles figure, a polytope constructed from those cycles of the quotient graph of the net that are associated with its geodesic lines.

Synthesis and characterization of terephthalate-intercalated NiAl layered double hydroxides with high Al content.

Terephthalate-intercalated nickel-aluminum layered double hydroxides (LDHs) were prepared by a co-precipitation method, with nominal x values in the general formula Ni((1-x))Al(x)(OH)(2)(C(8)H(4)O(4))(x/2) in the range 0.3-0.8.

Non-crystallographic nets with freely acting, non-abelian local automorphism groups.

Non-crystallographic (NC) nets are defined as periodic nets whose automorphism groups are not isomorphic to any isometry group in Euclidean space. This work focuses on a simple class of NC nets, restricted to nets with non-abelian, freely acting local automorphism groups. A general method is presented to derive such NC nets from crystallographic nets and some non-trivial examples are explored.

Synthesis and structural characterization of a new nanoporous-like Keggin heteropolyanion salt: K3(H2O)4[H2SiVW11O40](H2O)8+x.

Single crystals of the potassium salt K3(H2O)4[H2SiVW11O40](H2O)8+x of the vanadium monosubstituted alpha-Keggin dodecatunsgstosilicate were grown from an aqueous solution and analyzed by EDS, XRD, vibration and electronic spectroscopy, and 1H, 51V, and 29Si solid-state NMR spectroscopy. Results indicate the formation of a nanoporous-like compound of hexagonal symmetry (space group P62) with large, water-filled channels running along the c axis. A uniform distribution of vanadium over the 12 metal sites of the alpha-Keggin anion is observed by XRD.

Infinite geodesic paths and fibers, new topological invariants in periodic graphs.

Rings are well known invariants of nets. In this work, a generalization of the concepts of cycles and rings is introduced. Infinite paths in periodic graphs are defined as connected, acyclic, regular subgraphs of degree two; geodesics are defined as infinite paths such that the unique path between any pair of vertices is a geodesic path in the whole graph.

Characterization by 27Al NMR, X-ray absorption spectroscopy, and density functional theory techniques of the species responsible for benzene hydrogenation in Y zeolite-supported carburized molybdenum catalysts.

Carburized molybdenum catalysts supported on a dealuminated NaH-Y zeolite were prepared by carburization under a 20% methane in hydrogen flow of two precursors obtained by adsorption of molybdenum hexacarbonyl, one containing 5 wt % and the other 10 wt % Mo, and a third one was prepared by impregnation with aqueous ammonium heptamolybdate, containing 5 wt % Mo. The three catalysts displayed very distinct behaviors in the benzene hydrogenation reaction at atmospheric pressure and 363 K. By using XANES spectroscopy at the molybdenum L edge, EXAFS and XANES spectroscopy at the molybdenum K edge, and 27Al solid-state NMR spectroscopy, it was shown that different carburized molybdenum species exist in each sample.

A structural analysis of lead hydroxyvanadinite.

Hydroxyvanadinite, Pb(10)(VO(4))(6)(OH)(2), was prepared by the co-precipitation method and analyzed by X-ray absorption spectroscopy (XANES, EXAFS), infrared spectroscopy, Raman scattering and X-ray diffraction (XRD). The results showed that the structure is very similar to that of vanadinite, Pb(10)(VO(4))(6)Cl(2), with space group P6(3)/m (176) and cell parameters a = 10.2242(3) A and c = 7.

Topological density of nets: a direct calculation.

The topological density of a three-dimensional net is currently available after determining its entire coordination sequence. This paper shows that its value can be obtained directly from the set of cycles of the quotient graph of the net. A geometrical tool, the cycles figure of the net has been developed for this purpose.

Algebraic determination of generating functions for coordination sequences in crystal structures.

Given a connected crystalline structure, the set of paths and cycles of the quotient graph of its bond net is embedded into a commutative ring structure. Multiplication combines walks to build up geodesics of the net whereas addition stands symbolically for enumerating a collection of walks. Topological criteria are used to define zero divisors which enable the development of an algebraic generator into a combination of geodesics that are in a one-to-one correspondence with the vertices of the net.